Particle Swarm Optimization for entropy-based risk measures in portfolio selection problems: a mean – Entropic-VaR application

(1)

Master’s Degree Programme

Second Cycle (D.M. 270/2004)

in Economics and Finance - Finance

Final Thesis

Particle Swarm Optimization for

entropy-based risk measures in

portfolio selection problems:

A mean

– Entropic-VaR application

Supervisor

Ch. Prof. Marco Corazza Graduand

Riccardo Belli 871927

Academic Year

(2)

(3)

3

Introduction

Portfolio selection is a cornerstone of finance and economics. The problem consists in the minimization of a risk measure, while taking into account a series of constraints. Portfolio selection approach was introduced by Markowitz in 1952. His model was the first one to face the problem of how to efficiently invest a given amount of capital. Known also as Modern Portfolio Theory, Markowitz model revolutionized financial market investments. However, this model does present some limits and a set of assumptions rather utopic in the real world.

In this dissertation we will create a portfolio model trying to include some of the aspects that were avoided at that time, as for instance the presence of transaction costs and the allowance to buy or sell only determined quantities of assets. Furthermore, some model’s assumptions will be modified, in particular the concept of risk measure. According to the most recent literature, in fact, only the so-called coherent risk measures can be employed as real financial risk measures. The most common risk measures, that have been used as alternatives to variance, are Value-at-Risk (VaR) – even though not being coherent – and Expected Shortfall

(ES) or Conditional Value-at-Risk (CVaR).

The risk measure chosen for the portfolio model developed in this work belongs to the class of entropy-based risk measures, is a coherent risk measure introduced by Ahmadi-Javid1_{and is called Entropic Value at Risk (EVaR). All the}

above-mentioned measures of risk will be accurately described and discussed later in the dissertation.

Conversely to what proposed by Markowitz, the model developed in this work will not be based on the mean-variance criterion, but rather on mean-entropic VaR. Moreover, a system of characteristic Markowitz model’s constraints will be

1_{“Entropic Value-at-Risk: A New Coherent Risk Measure”, A. Ahmadi-Javid (See}

(4)

4

applied, as budget constraint and desired minimum return. We will successively introduce mixed-integer constraints, useful for managing transaction costs. Since obtaining exact results from a constrained minimization problem is highly time-consuming, the solution proposed in the present dissertation will be metaheuristic-based (the concept of metaheuristic will be discussed in what follows): the metaheuristic employed, Particle Swarm Optimization, will not give an exact result to the problem, but a good level of approximation. This method consists in the employment of a bio-inspired metaheuristic algorithm able to search for an optimal solution to the problem, while not exact, exploiting the dynamics of exploration of groups of animals in the nature, like birds’ flocks or shoals of fish.

The application of the model will be performed by Matlab and the results will be compared with the application in mean-Expected Shortfall of the same portfolio selection model.

(5)

5

Chapter 1

Portfolio selection problem

As well explained by (Constantinides G.M. and Malliaris A.G. 1995), in general a consumer, given a certain amount of income, typically faces two important economic decisions: the first one consists in deciding how to allocate his or her consumption among goods or services; the second one is the decision on how to invest among various assets. These two interrelated problems are known as the

consumption-saving decision and the portfolio selection problem.

Portfolio selection is one of the most discussed and interesting problems in the economics and finance world. Modern portfolio theory finds his pioneer in Harry Markowitz, which developed the Mean-Variance portfolio selection model in 1952. Despite being recognized as one of the cornerstones in the portfolio selection problem, Markowitz’s model proved to be too simplistic to represent the actual real world and its basic assumptions have been widely contested in recent years.

In this chapter we will give a synthetic but complete description of the Mean-Variance portfolio selection model by Harry Markowitz and a brief description of some of the models that try to overcome its limits. We will then shift our interest on the desirable characteristics for a risk measure, closing the chapter focusing on the importance of the adoption of a coherent risk measure in this environment.

(9)

9

1.1 Markowitz Model

Almost seventy years ago American economist Harry Markowitz developed a model which revolutionized investment the practise and became in the course of time one of the pillars of financial economics and Modern Portfolio Theory. His Mean-Variance model, rewarded with the Nobel Prize in Economics in 1990, aims at selecting a group of assets which have collectively lower risk than any single asset on its own.

1.2 Basic assumptions

As said previously, the mean-variance analysis has been challenged through the years due to the simplicity of the model with respect to the real world. The model limitations are given by some strong assumptions on which it relies:

• Investors always maximize the rate of return yielded by their investments;

• Investors are rational and risk-averse2_{: they are completely aware of all the}

risk underlying an investment and take positions basing their decisions on the risk, asking higher returns for accepting higher risk and coherently expecting lower return for lower levels of risk;

• Investors make their investment judgements by taking into consideration expected returns and standard deviation (risk measure) of returns of the possible assets;

• Investments have a single period horizon, meaning that at the beginning of the period t the investor allocates her/his wealth among different assets

2_{If faced by the decision between two identical portfolios, a risk-averse investor will}

(10)

10

and she/he will hold the portfolio until the period t + ∆t, without considering the opportunity to reinvest the wealth in a following period;

• Under the condition of uncertainty, investors - and more in general, individuals - make decisions by maximizing the expected value of an utility function of consumption, which is assumed to be increasing and concave;

• Investors’ assets are infinitely divisible. Thus, investors may decide to buy or sell a fraction of a share;

• Investors are price-takers, meaning their actions can’t affect the probability distributions of returns on the available securities;

• Financial markets are frictionless. Hence, there are no transaction costs, no taxes, absence of institutional restrictions, and so on.

1.3 Markowitz portfolio selection model

The portfolio selection process may be seen as constituted by three stages:

1. The first stage consists in the identification of appropriate measures for measuring the expected return and risk;

2. The second stage establishes a criterion to identify the “best” portfolios, distinguishing between efficient portfolios and non-efficient ones;

(11)

11

3. In the third and last stage takes place the selection of a proper portfolio for the investor, according to her/his risk aversion. This activity is pursued by maximizing the investor’s expected utility function.

1.3.1 Measures of Risk and Return

The future profitability of an asset is uncertain at the time of the purchase. This uncertainty is given by the randomness associated to return, since we do not know the future price ex-ante.

There are some statistical tools which help the investor to manage face the uncertainty of the investment:

• The mean of the single-period rate of return. It represents the profitability – expected return – of an investment;

• The variance of the single-period rate of return. It represents the risk of an investment and it is of course undesirable;

• The correlation between the return of each pair of risky assets. It represents the linear dependency between the pair of returns of the assets.

Expected value (mean) and variance of individual securities returns can be defined as follows:

Let 𝑋 be a discrete random variable 𝑋 = {(𝑥1, 𝑝1), … , (𝑥𝑖, 𝑝𝑖), … , (𝑥𝑀, 𝑝𝑀)}, where

𝑥_𝑖 with 𝑖 = 1, … , 𝑀 is the possible return from a given asset, and 𝑝𝑖, with 𝑖 =

1, … , 𝑀, is the probability of occurrence of 𝑥𝑖, with 0 ≤ 𝑝𝑖 ≤ 1 for all 𝑖 and

(12)

12 𝐸(𝑋) = ∑ 𝑥_𝑖𝑝_𝑖 𝑀 𝑖=1 𝑉𝑎𝑟(𝑋) = ∑(𝑥_𝑖− 𝐸(𝑥))2𝑝_𝑖 𝑀 𝑖=1

Since a portfolio is a set of two or more individual securities, we can now define its expected rate of return and its variance. Let 𝑅𝑝 be the portfolio rate of return:

𝑅_𝑝 = 𝑥₁𝑅₁+ ⋯ + 𝑥_𝑁𝑅_𝑁 = ∑ 𝑥_𝑖𝑅_𝑖

𝑁

𝑖=1

where 𝑅𝑖 is the random variable representing the return of the 𝑖-th asset and 𝑥𝑖

the portion of capital in percentage invested on the same asset. Let 𝑟𝑖 and 𝜎𝑖2 be

respectively the expected rate of return and the variance of the 𝑖-th asset, with 𝑖 = 1, … , 𝑁. Then the portfolio expected return and variance can be defined as follows: 𝐸(𝑅_𝑃) = ∑ 𝑥_𝑖𝑟_𝑖 𝑁 𝑖=1 ≔ 𝑟_𝑃 𝑉𝑎𝑟(𝑅_𝑃) = ∑ 𝑥_𝑖2_𝜎 𝑖2 𝑁 𝑖=1 + 2 ∑ ∑ 𝑥_𝑖𝑥_𝑗𝜎_𝑖,𝑗 𝑁 𝑗=𝑖+1 𝑁 𝑖=1 = ∑ 𝑥_𝑖2𝜎_𝑖2 𝑁 𝑖=1 + 2 ∑ ∑ 𝑥_𝑖𝑥_𝑗𝜌_𝑖,𝑗𝜎_𝑖𝜎_𝑗 𝑁 𝑗=𝑖+1 𝑁 𝑖=1 : = 𝜎_𝑃2

(13)

13

where 𝜎𝑖,𝑗 = 𝜌𝑖,𝑗𝜎𝑖𝜎𝑗 is the covariance and 𝜌𝑖,𝑗∈ [−1, 1] is the linear correlation

coefficient between 𝑅𝑖 and 𝑅𝑗.3

Mean and variance of the portfolio might also be defined with the use of vectorial notation:

o 𝑟_𝑃 = 𝑥′_𝑟;

o 𝜎_𝑃2 _{= 𝑥}′_𝑉𝑥.

where 𝑉 is the usual variance-covariance matrix.

1.3.2 Mean-Variance Dominance Criterion

The efficiency criterion proposed by Markowitz for the second stage is the Mean-Variance Dominance Criterion. It takes into consideration mean, that is the expected return, and variance to differentiate between efficient portfolios and inefficient ones, leading the investor to seek the lowest variance for a given expected return or the highest expected return for a given level of variance.

Definition. Mean-Variance Dominance Criterion. Given two random

variables 𝑋 and 𝑌, respectively with mean 𝜇_𝑋 and 𝜇_𝑌 and variance 𝜎_𝑋2 and 𝜎_𝑌2, it is possible to state that 𝑋 dominates 𝑌 with respect to the mean-variance criterion if and only if the following three conditions hold simultaneously:

1. 𝜇𝑋 ≥ 𝜇𝑌 ;

2. 𝜎𝑋2 ≤ 𝜎𝑌2 ;

3. At least one of the previous inequalities is verified in narrower sense.

3_{If 𝜌 > 0 the return of the two assets move in the same direction and are positive}

correlated; if 𝜌 < 0 the return move in the opposite direction and they are negatively correlated; if 𝜌 = 0 there is no relationship between the two assets.

(14)

14

It is clear that the criterion introduces a partial basis for comparison since it does not allow to discriminate all pairs of portfolios. In fact, it may happen for instance that when considering two efficient portfolios, we are not able to determine whether one portfolio dominates the other.

It is possible to state that the set of efficient portfolios, named efficient frontier, is constituted by all portfolios which, alternatively, once determined the desired level of expected return, minimize portfolio’s risk or, given a certain level of risk, maximize the expected return. Rational investors’ choice can fall only on a portfolio belonging to this set whereas for every inefficient portfolio there is one which, carrying the same risk, can guarantee a greater return or, equivalently, having the same expected return, guarantees a lower risk.

In the case of N assets with random returns, Markowitz formulation for the portfolio selection problem can be stated as follows4_:

minimize 𝑥 𝑥 ′ 𝑉𝑥 subject to { 𝑥′_{𝑟̅ = 𝜋} 𝑥′_{𝑒 = 1} 𝑥 ≥ 0 where:

• 𝑥 is the N-order vector constituted by the portion of wealth 𝑥1, … , 𝑥𝑛

invested in the 𝑖-th asset of the portfolio, with 𝑖 = 1, 2, … , 𝑛; • 𝑉 is the N-order quadratic matrix of variances and covariances5_;

4_{This formulation is an example of quadratic program, an optimization problem}

constituted by a quadratic function and linear constraints.

5_{Given the symmetric nature of covariances, the matrix is as well symmetric by definition,}

with variances on its diagonal. We assume that the matrix is non-singular: none of the assets returns is perfectly correlated with the return of a portfolio composed by the remaining assets and none of the assets is riskless.

(15)

15

• 𝑟̅ is the N-order vector composed by mean returns 𝑟1, … , 𝑟𝑁 of N assets6;

• 𝑒 is a N-order unitary vector;

• 𝜋 is the level of expected return that the investor wishes.

The constraints considered by Markowitz are basic and they can be explained as follows: the first one implicates that, in the process of risk minimization, the level of expected return desired by the investor and fixed ex-ante 𝜋 must be taken into consideration; the second constraint requires that the entire wealth at disposal is invested; the last ones imply that the portions of wealth invested in each asset are non-negative, in order to avoid short selling7_.

With the aim of determining a unique vector of optimal weights, the following statement should be made.

Theorem. If the variance-covariance matrix 𝑉 is positive definite8_{and non-singular}

– hence invertible – and if there is at least one pair of different mean returns, then the optimization problem admits a unique solution.

Notice that the first two linear constraints define a convex set and, being 𝑉 positive definite, also the function 𝑥′𝑉𝑥 is convex.

To find the formula for the optimal portfolio given the constraints, we shall start from the following lagrangian function:

𝐿 = 𝑥′_{𝑉𝑥 − 𝜆}

1(𝑥′𝑟 − 𝜋) − 𝜆2(𝑥′𝑒 − 1)

6_{It is assumed that not all elements of 𝑟 are equal. Conversely, the entire wealth would}

be invested in the asset with the lowest variance.

7_{Short selling is a particular investment or trading strategy which involve the sale of a}

security not owned by the seller. It is undertaken when the seller has the belief that the price of the security will decline or when an investor wants to hedge, placing an offsetting position to reduce risk exposure.

(16)

16

where 𝜆1 and 𝜆2 are the Lagrange multipliers.

We can now set equal to zero the partial first derivatives of 𝐿 and set up the system: { 𝜕𝐿 𝜕𝑥 = 2𝑥 ′_{𝑉 − 𝜆} 1𝑟′− 𝜆2𝑒′= 0 𝜕𝐿 𝜕𝜆1 = −𝑥′𝑟 + 𝜋 = 0 𝜕𝐿 𝜕𝜆₂ = −𝑥 ′_{𝑒 + 1 = 0}

With some computations it is possible to obtain the final unique solution to optimization problem: 𝑥 = (𝛾𝑉 −1_{𝑟 − 𝛽𝑉}−1_{𝑒)𝜋 + (𝛼𝑉}−1_{𝑒 − 𝛽𝑉}−1_𝑟) 𝛼𝛾 − 𝛽2 where: 𝛼 = 𝑟′𝑉−1𝑟 𝛽 = 𝑟′𝑉−1𝑒 = 𝑒′𝑉−1𝑟 𝛾 = 𝑒′𝑉−1𝑒

The efficient frontier’s analytical expression varies depending on the composition of the portfolio:

➢ Portfolio with N > 2 risky assets: as in the case explained, the frontier expression is still represented by a parabola in the mean-variance plane,

(17)

17

but the vertex changes depending whether a risk-free asset is considered or not.

➢ Portfolio with N = 2 risky assets: in this scenario the frontier expression is particularly affected not only by the possible presence of a risk-free asset, but also by the linear correlation coefficient between the two assets.

1.3.3 Portfolio Selection

In the third and last stage the proper portfolio for the investor is selected, taking into consideration the investor’s risk aversion and knowing that usually all investors prefer returns to be high and/or stable, not subject to uncertainty. In order to do so, we consider the investor’s expected utility function and we maximize it, seeking to obtain from one of the portfolios laying on the efficient frontier the greatest utility for the investor.

In his model, Markowitz adopted the quadratic utility function, described by the following equation: 𝑈(𝑅𝑃) = 𝑅𝑃− 𝑎 2𝑅𝑃 2 where:

• 𝑅_𝑃 is the random variable representing the return of the portfolio;

• 𝑎 is a strictly positive coefficient reflecting the investor’s risk aversion: the greater is 𝑎, the greater is the investor’s risk aversion.

However, the compatibility between mean-variance criterion and the theory of expected utility maximization occurs only in two limit cases: following a “subjective” approach, when the utility function of all investors has quadratic form, whereas, following an “objective” approach, when the joined probability

(18)

18

distribution function of the N assets constituting the portfolio is a multivariate elliptical one, independently from the utility function form.

Hence, it is necessary that the efficient frontier is consistent with the maximization of the expected utility. Knowing the diversity of forms that can characterize the efficient frontier, such as the variety of possible values that the linear correlation coefficient can assume, the determination of the optimal portfolio for a specific investor can be formulated as the following constrained maximization problem:

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝐸[𝑈(𝑅𝑃)]

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝜎_𝑃2 _{= 𝑓(𝑅} 𝑃)

Concerning this scenario, one of Markowitz’s most significant contribution can be considered the concept of diversification in the financial world.

Definition. Diversification. There is diversification when, allocating wealth at

investor’s disposal in N > 1 assets, it is possible to exploit the correlation between the assets to reduce portfolio risk under the level of the portfolio’s asset with the lowest risk.

Recalling the mathematical properties of the variance, regarding the sum of non-independent random variables9_{, it is possible to infer the key role played by the}

linear correlation coefficient. In fact, albeit in the reality of financial markets the existence of assets linked together by a linear correlation coefficient of -1 is unlikely, it is perceivable how also coefficients with less extreme values can participate in the decrease of overall portfolio risk.

9_{Given two random variables 𝑋 and 𝑌, the variance of their sum can be defined as 𝜎}

𝑋+𝑌2 =

𝜎𝑋2+ 𝜎𝑌2+ 2𝜎𝑋,𝑌 = 𝜎𝑋2+ 𝜎𝑌2+ 2𝜌𝑋,𝑌𝜎𝑋𝜎𝑌, with 𝜎𝑋,𝑌 being their covariance, 𝜎𝑋 and 𝜎𝑌

being respectively the standard deviation of 𝑋 and 𝑌 and 𝜌𝑋,𝑌 being the correlation

(19)

19

Referring to the portfolio overall mean return, being a linear combination of its single assets’ expected returns, in absence of short selling – as in the case previously presented – it has to be included between the lowest and the highest of the single assets’ mean returns. Whereas, with the exploitation of short selling it is possible to enlarge the invested capital with respect to the initial quantity. In terms of portfolio returns this leads to the chance of obtaining a higher return in mean in relation to the portfolio’s asset with the highest return. In this case of course the greater expected return will come at the price of a higher overall portfolio risk.

The limit of Markowitz model’s low diversification is that, since in the real world empiric correlations between assets are rather low, to reduce the overall variance the mean-variance criterion often tend to over-weight assets with low variances instead of exploiting negative relations between their market price trends.

1.4 Critics to Markowitz Model

As mentioned earlier, Markowitz model constitutes a fundamental contribution to the present quantitative finance on the subject of portfolio theory. However, the assumptions underlying the model which nowadays appear rather simplistic, were more suitable at the half of the last century, period in which the model was first presented and applied. Hence it seems appropriate discussing the main limits faced by the model, concerning its basic assumptions and the type of constraints employed.

Returns distribution:

The first limit regards hypothesis underlying returns theory. The Normal density probability function generally assumed poorly describes financial assets returns in the real market. In fact, a general tendency observed in assets returns is negative asymmetry in returns distribution. A Normal distribution variable is characterized by an asymmetry equal to zero. On the other hand, asymmetry suggests that a distribution has its barycentre towards values greater than the mean, in which

(20)

20

case asymmetry value will be positive, or towards values lower than the mean, in which case asymmetry value will be negative. A situation of negative asymmetry in returns distribution reveals that negative events often lead to greater negative returns in proportion with respect to positive returns given by favourable events. This scenario can be observed for instance in the case of bonds, where the consequence of a default10_{lead to a great negative return, while positive return is}

fixed.

Additional consideration that increases differences between real market and the Normal distribution assumption is kurtosis11_{. It has been shown that returns}

distributions observed on the market have greater values of kurtosis. Hence, extreme events (very positive or very negative returns) are more frequent with respect to theoretical returns with Normal distribution.

Expected utility function:

The assumption of a quadratic utility function has been often criticized. In particular, following the most common economic principles, as wealth increases also utility should increase, while it is evident that a quadratic utility function decreases above a certain level of wealth. This implication is of course controversial with the hypothesis of non-satiety of investors. Furthermore, other critics show that the implicit theoretical weakness in the assumption of utility function’s quadratic form lies in the increasing risk aversion deriving from it. On the contrary, economic theory suggests that a decreasing risk aversion corresponding to the increase of wealth would be more suitable with economic agents’ behaviour.

Measure of risk:

Being a symmetric measure, the use of variance as a risk measure has been as well disapproved since it considers positive and negative returns12_{equally weighted.}

10_{Complete or partial payment of capital and interests not fulfilled.}

11_{Kurtosis is an index relative to distribution’s form. It measures the thickness of tails: a}

Normal distribution has a kurtosis value of 3, if tails are fatter the value is bigger than 3 and vice versa.

12_{In this context, positive and negative returns can also be referred as upside potential}

(21)

21

However, it has been shown that investors do not treat them with the same weight. The utilization of variance as a risk measure can only be consistent in case returns present a symmetric distribution. This aspect was noticed from Markowitz himself few years after the release of his article, as we will discuss in the following paragraph.

Other assumptions:

Additional unrealistic assumptions contribute to constitute more practical limits to Markowitz model, such as:

• Friction-less market: absence of transaction costs or taxation costs. They affect portfolio performance; thus, they assume a major role in portfolio management;

• Absence of constraints deriving from the economic-politic context in which one is operating;

• Absence of constraints relative to the possibility of buying or selling an asset in a finite number.

Despite being questioned for many assumptions and limits, Markowitz model remained the most important contribution to Modern Portfolio Theory. This relevance, in conjunction with continuous debating, has led to various improvements of the model.

1.5 Improvements of Markowitz Model

With the aim of making the model proposed by Markowitz more realistic, it is possible to follow two paths. First, constraints system can be modified in order to make it more consistent with the real world; we could add for instance market friction limits as transaction and taxation costs or take into account mixed-integer

constraints. Secondly, it is possible to operate on the objective function, hence on

(22)

22

In this work, the attention will be prevalently addressed to the objective function and its risk measure; nonetheless, it seems useful in author’s opinion to provide a brief description of mixed-integer constraints, that help portfolio selection problem to gain a more realistic form.

Mixed-integer constraints can be divided into three categories:

▪ Constraints relative to transaction’s minimum lots, which have to be negotiated only in an integer number of units;

▪ Constraints relative to the maximum positive integer number of different assets which can be negotiated;

▪ Constraints relative to the minimum positive integer number of minimum lots of a given asset which has to be negotiated.

On one hand, the introduction of these limits provides a greater computational investigation of the programming problem; while on the other hand, it notably increases the complexity of the solution process. Indeed, the observance of mixed-integer constraints in a mathematical programming problem constitutes a problem known as NP-complete13_{and moreover, solving such mathematical}

programming problem is a NP-hard14_problem.

As mentioned earlier, for the purpose of our work, we will undertake the second path which involves the objective function.

Since the beginning the use of returns’ variance as risk measure inherent an investment portfolio appeared misleading. It was in fact Markowitz himself that proposed in 1959 its substitution in favour of semi-variance, formulated as follows:

13_{NP-complete are problems which are considerably burdensome to solve in terms of}

time requirement. In computational complexity theory, NP-complete are the most difficult problems in the NP class (non-deterministic polynomial-time problems).

14_{hard are problems that are difficult at least as much as (or not less than)}

(23)

23 𝑆𝑒𝑚𝑖 − 𝑣𝑎𝑟 (𝑅𝑃) = 1 𝑁∑ (𝑅𝑖− 𝜇) 2 𝑁 𝑖=1;𝑅_𝑖<𝜇 .

While adopting this alternative, only potential losses are considered, thus returns realizations which are below their expected value. Any rational investor would not disdain to own a portfolio with returns above the expectations – hence the expected value –, so the notion of risk incorporated in variance definition does not result appropriate in this scenario. When a portfolio yields more than the expected return, it is suitable to refer to it as an opportunity, not a risk. Distinction between downside risk and upside potential appeared necessary in order to conceive risk as an element essentially negative rather than a simple dispersion measure. Following the previous analysis, it seemed indispensable providing a definition of risk measure and identifying certain desirable characteristics.

1.5.1 Defining a risk measure

Although it is feasible to define certain desirable characteristics that a risk measure should have, the concept of risk is rather subjective15_{and it is, thus, not}

possible to univocally identify a measure capable of satisfying the problem of expected utility maximization – common to every investor – once the efficient frontier is determined. However, even though being affected by relativity and subjectivity, it is possible to determine some characteristics that a risk measure should have relatively to a specific set of investors: rational agents.

In order to find a risk measure that satisfies properly investors’ preferences, different methodologies have been undertaken. Accordingly, a risk measure can be described with the following definition:

Definition. Risk Measure. A risk measure is a function 𝜌 that assigns a

non-negative numeric value to a random variable 𝑋, which can be interpreted as future return, 𝜌: 𝑋 → 𝑅.

(24)

24

It is possible to identify the basic properties that a risk measure should satisfy in order to define function 𝜌 as aforementioned, even though it is not enough to take into consideration. In fact, the concept of coherent risk measure is greatly relevant nowadays and will be discussed in the following paragraph.

We begin from the relevant and desirable characteristics that a risk measure should have:

➢ Positivity: a measure of risk associated to a random variable assumes a strictly positive value, at least null in case there is no randomness. Negative values do not make sense;

➢ Linearity: especially in the resolution of optimization problems of big dimensions, the computational complexity might be diminished linearly linking risk measure and future return. The goodness of certain risk measures is connected to the more treatable computations that come from a linear optimization problem, where risk and return are linked in a basic way;

➢ Convexity: a risk measure is convex if, given two random returns 𝑅𝑋 and

𝑅_𝑌 and a parameter 𝜗 ∈ [0; 1], the following relation holds:

𝜌(𝜗𝑅_𝑋+ (1 − 𝜗)𝑅_𝑌) ≤ 𝜗𝜌(𝑅_𝑋) + (1 − 𝜗)𝜌(𝑅_𝑌).

It is a property that highlights the importance of diversification, since it is a process that permits to reduce the overall portfolio risk, hence, to expose invested wealth to a minor risk.

This property can be satisfied indirectly, satisfying the two following properties:

▪ Subadditivity: 𝜌(𝑅𝑋+ 𝑅𝑌) ≤ 𝜌(𝑅𝑋) + 𝜌(𝑅𝑌);

(25)

25

The procedure of risk measure minimization, given a certain level of expected return, has the aim of bounding the uncertainty linked to the invested capital future value rather than the increment of the latter, taking into consideration that the expected return level is established. In this way, it is possible to estimate in advance what the invested capital future value will be, with a level of uncertainty depending on returns’ distribution variance. Consequently, the resulting portfolio can be defined optimal only by a risk-averse investor, which cannot consider also the concept of non-satiety.

Gathering in a unique real positive number all probability distribution’s characteristics, it is clear that an important limit of risk measures is constituted by their incapacity of incorporating the whole information available in a stochastic order, which utilizes the losses’ cumulative distribution function.

An important measure of downside risk that has been widely used in the economic and financial environment is Value at Risk (from now on, VaR). Its notion is rather simple and intuitive:

Definition. Value-at-Risk. Given a confidence level of 𝛼 ∈ [0; 1] and fixed a

specific holding period, Value at Risk (VaR) indicates the maximum potential loss associated to a portfolio in 𝛼% of cases during the holding period.

Originally conceived as synthetic indicator of market risk16_{, this measure of risk is}

notably widespread in savings and credit industries. Albeit having an experienced creator – it was conceived within the American investment bank J.P. Morgan – and expressing risk in the same unit measure of invested capital (monetary terms), VaR presents various limits when the underlying losses are not distributed as a Normal. Even in this case however, the assumption of Normality – rather widespread in literature – on one hand permits to lead back the portfolio optimization problem based on VaR to a Markowitz approach, while on the other hand leads to a not

16_{Market risk is defined as risk linked to adverse movements in financial activities’ prices,}

(26)

26

negligible underestimate of portfolio’s real VaR, since returns of portfolios containing derivatives or tools to which is associated a low rating tend to have a strongly left-asymmetric distribution (negative asymmetry).

Regarding the computational profile, this measure does not fit in a particularly convenient way to bounded optimization problems since it emerges a stochastic programming problem rather difficult to solve. Moreover, with the exception of the case in which the underlying positions’ probability distributions are known, it is complex to obtain a precise measure of portfolio’s VaR.

From a probabilistic standpoint, VaR with a confidence level of 𝛼 is the value that satisfies the following equality:

𝑃(𝐿 > 𝑉𝑎𝑅𝛼) = 1 − 𝛼

Where 𝐿 is a generic distribution of losses. Complying with this interpretation, it is evident how being a threshold measure17_{– indeed it express the maximum}

potential loss with a certain level of probability – does not provide any indication on the size of losses that exceeds that threshold, thus on the nature of the profit and loss distribution’s left tail (the portion exceeding VaR). The distortion presented tends to be towards lower losses, leading to a contrast with the theory of risk management, which privileges a more cautious and pessimistic behaviour in the determination of risk level associated to a portfolio.

Lastly, even though not being less relevant, another gap presented by VaR concerns the aggregation of more risk sources. The above-mentioned measure does not encourage – sometimes even prohibits – diversification, since it does not take into account events’ potential economic consequences. With such behaviour, VaR does not satisfy the feature of subadditivity – a property that will be seen more in detail in the next paragraph – since, applying this measure, the overall portfolio risk could result even greater than the sum of the single risk sources underlying each asset. It is important to underline how the lack of subadditivity creates, in addition to the inconsistency with the diversification principle, issues

(27)

27

with the numeric treatability. In fact, VaR is also criticized for is inability to quantify the so-called tail risk, hence, its low sensibility to extreme events.

1.5.2 Concept of coherent risk measure

In order to complete the process of individuation of desirable properties that a risk measure should have and to provide a follow-up to the inadequacies of VaR measure, it is appropriate to describe the concept of coherence, as formulated by Artzner et al. (1999).

The writing of coherence axioms represented an attempt of translating a complex reality into a mathematical formulation that is not so restrictive to identify a unique coherent measure of risk, but it rather characterizes a class of measures. As mentioned above, in addition to the “basic” properties of a risk measure, there are other significant properties. The respect of these additional features is a necessary condition to a correct interpretation of the concept of risk associated to a financial instrument.

Definition. Coherent Risk Measure. A risk measure that satisfies the four

axioms of translation invariance, subadditivity, positive homogeneity and

monotonicity is called coherent.

We can now list and describe the four properties that define a coherent risk measure:

• Translation invariance: it guarantees that investing a percentage 𝛼 of the available capital in a risk-free asset18_{, the overall risk associated to the}

portfolio contracts proportionally to the percentage 𝛼 allocated in the risk-free asset:

18_{Asset that has a known future return and that does not carry any level of risk. Usually}

(28)

28

𝜌(𝑋 + 𝛼) = 𝜌(𝑋) − 𝛼 , ∀ 𝑟. 𝑣. 𝑋 , 𝛼 ∈ ℝ.

It implies that 𝜌(𝑋 + 𝜌(𝑋)) = 0. By adding a risk-free quantity equal to 𝜌(𝑋) to a risky position 𝑋, we obtain a risk-free entity, coherently with the operative interpretation of 𝜌 as minimum positive quantity to add to the initial position in order to make the instrument acceptable (thus, risk-free); • Subadditivity: it represents the essence of how a risk measure should behave in the case in which the investor has to deal with a combination of assets. The risk of a portfolio should never be greater than the sum of the single risks associated to each of the assets that constitutes it. Subadditivity is strictly correlated to the concept of diversification since it can be affirmed that diversification leads to a contraction of the overall risk only if, for the risk inherent to a certain position, the following statement holds:

𝜌(𝑋 + 𝑌) ≤ 𝜌(𝑋) + 𝜌(𝑌) , ∀ 𝑟. 𝑣. 𝑋, 𝑌.

• Positive homogeneity: it ensures that if the investment in a risky asset varies, then the riskiness associated to that investment varies proportionally. In cases in which positions dimensions directly affect risk (e.g. if positions are so large that time required to liquidate them depends on their dimensions), consequences of lack of liquidity should be considered when calculating the future net worth of a position:

(29)

29

• Monotonicity: it underlines the preferability of an asset that systematically assures returns greater than another asset:

𝜌(𝑋) ≤ 𝜌(𝑌) , ∀ 𝑟. 𝑣. 𝑋, 𝑌 𝑤𝑖𝑡ℎ 𝑋 ≥ 𝑌.

As mentioned earlier, the concept of coherent risk measure does not define a unique risk measure, instead, it characterizes a large class of risk measures. The choice of the right measure to use within the class should be made based on some additional economic considerations.

First examples of coherent risk measures

In their “Coherent Measures of Risk”, where they defined the axioms of coherence, Artzner et al. (1999) provided also the first guidelines concerning some proposals of coherent risk measures that satisfy the axioms. They present two measures, known as Tail Conditional Expectation (TCE) and Worst Conditional Expectation

(WCE), for whom the authors demonstrated that the relation 𝑇𝐶𝐸𝛼 ≤ 𝑊𝐶𝐸𝛼19

holds.

Definition. Tail Conditional Expectation. Tail Conditional Expectation (known

also as TailVaR) is a coherent risk measure defined as: 𝑇𝐶𝐸𝛼(𝑋) ≝ −𝐸[𝑋|𝑋 ≤ −𝑉𝑎𝑅𝛼(𝑋)].

Definition. Worst Conditional Expectation. Worst Conditional Expectation is a

coherent risk measure defined as:

𝑊𝐶𝐸_𝛼(𝑋) ≝ − inf{𝐸[𝑋|𝐴] | 𝑃[𝐴] > 𝛼}.

(30)

30

Financially, TCE and WCE try to determine “how bad is bad” since they focus on returns distribution’s left tail – the one representing losses – and they compute the mean value subject to the fact that the losses are greater than a certain value. The concepts of TCE and WCE represent first proposals of coherent risk measures. However, at the same time, their resemblance could erroneously let their aspects of distinction pass unnoticed. If on one hand WCE satisfies completely axioms of coherence and, nonetheless, it is widespread only in the theoretical field – since it requires knowledge of the entire underlying probability space – , on the other hand TCE is more manageable also in the application environment, even though not always satisfying axioms of coherence20_.

Conversely, as discussed before, the adoption of VaR as risk measure does not provide any indication on the size of losses beyond a threshold value, represented by the measure itself. The introduction of the axioms of coherence leads to a change in the question that we can ask ourselves with the aim of determining a more suitable risk measure, hence, coherent. More specifically, risk measures discussed in this paragraph do not observe at the maximum potential loss in the 𝛼% of cases, but they rather point out the expected loss in the worst case (1 − 𝛼)% scenario. In other words, these measures do not concentrate on a specific threshold, which does not supply with any information besides the threshold itself, but they focus on the losses’ distribution beyond the threshold value and they synthetize its features through their mean value.

The aim of creating a measure of risk that combined contemporarily the good qualities of both measures was reached through the definition of an alternative and more suitable solution represented by the measure known as Expected

Shortfall (ES). This index can be financially explained as the average loss

considering all losses beyond a certain threshold value, VaR.

20_{TCE measure may not always respect the property of subadditivity. In fact, TCE}

coherence is guaranteed only restricting the analysis field on the continuous probability distribution’s functions, whereas it might not be guaranteed in the general case.

(31)

31

Expected Shortfall can be formally defined as follows.

Definition. Expected Shortfall. Given a profit and loss distribution 𝑋 and defined

holding period and significance level 𝛼 ∈ [0; 1], Expected Shortfall is defined as: 𝐸𝑆𝛼(𝑋) ≝ −

1

𝛼(𝐸[𝑋1(𝑋≤𝑥𝛼)] − 𝑥

𝛼_{[𝑃[𝑋 ≤ 𝑥}𝛼_{] − 𝛼])}

where 𝑥𝛼 = 𝑉𝑎𝑅.

The second addendum of the sum within the parenthesis can be translated as the quantity to subtract from the mean value when 𝑋 ≤ 𝑥𝛼 has probability greater than 1 − 𝛼. Whereas, when 𝑃(𝑋 ≤ 𝑥𝛼) = 1 − 𝛼, as it is usually the case with probability distribution’s continuous functions, we obtain that the value resulting from the ESα’s formula coincides with the TCEα’s one.

An equivalent representation that provide the advantage of more transparency and that permits to appreciate the simplicity of ES, can be obtained renouncing to the definition in terms of expected values. Let 𝐹(𝑋) be the probability density function21_{so that 𝑃(𝑋 ≤ 𝑥) and let 𝐹}−1_{(𝛼) = inf {𝑥|𝐹(𝑥) ≥ 𝛼} be the inverse}

function of 𝐹(𝑋), it can be proved that ES can be expressed as 𝐸𝑆𝛼(𝑋) =

−1

𝛼∫ 𝐹

−1_{(𝑝)𝑑𝑝} 𝛼

0 .

The sample estimation of ES is obtained sorting the 𝑛 possible realizations and, given a significance level, selecting the (1 − 𝛼)% of the greater losses and obtaining the following result: 𝐸𝑆𝛼(𝑋) = −

∑𝑤𝑖=1𝑥1:𝑛

𝑤 , where w represents the

integer part of 𝑛(1 − 𝛼)%, hence 𝑤 = max{𝑚|𝑚 ≤ 𝑛(1 − 𝛼), 𝑚 ∈ ℕ}.

ES is a universal risk measure, meaning that is applicable to any financial tool and to any underlying risk source. Moreover, it benefits of simplicity and completeness properties since it computes a unique number even in case of portfolios exposed to different risk sources and robustness. This is possible because, conversely to other risk measures focusing on distributions’ tail, with ES, results do not vary

21_{Probability density function of a random variable X is a non-negative application 𝑝}

𝑥(𝑥)

so that the probability of a set A is given by 𝑃(𝑋 ∈ 𝐴) = ∫ 𝑝_𝐴 𝑥(𝑥)𝑑𝑥 for all subsets A of

(32)

32

significantly when changing the confidence level of some point basis. This last aspect cannot be guaranteed by VaR, TCE or WCE.

An alternative expression of ES is the one proposed by Rockafellar and Uryasev (2002), named Conditional Value-at-Risk (CVaR). Let the function associated to the loss be 𝑧 = 𝑓(𝑥, 𝑦)22_{with Ψ(𝑥, 𝜁) = 𝑃{𝑦|𝑓(𝑥, 𝑦) ≤ 𝜁}, CVaR can be defined as}

follows:

Definition. Conditional Value-at-Risk. Fixed a significance level of 𝛼 ∈ [0; 1],

CVaRα is equal to the expected value of the greater losses whose probability is

equal to 1 − 𝛼. It is equivalent to the average of the distribution function:

Ψ_α(𝑥, 𝜁) = { 0, 𝑖𝑓 𝜁 < 𝜁𝛼(𝑥) [Ψ(𝑥, 𝜁) − 𝛼]/[1 − 𝛼], 𝑖𝑓 𝜁 ≥ 𝜁_𝛼(𝑥)

where 𝜁𝛼(𝑥) is the VaRα associated to portfolio 𝑥.

Rockafellar and Uryasev (2000) themselves, besides demonstrating its coherence23_{, highlighted an interesting further aspect: solving a simple convex}

optimization problem it is feasible to obtain separately both CVaRα and VaRα

associated to portfolio 𝑥. It is a result of particular importance since it allows to compute CVaRα of a position without necessarily knowing the relative VaRα. Both

risk measures can be determined simultaneously exploiting the following formula:

Fα(𝑥, 𝜁) = 𝜁 +

1

1 − 𝛼𝐸{[𝑓(𝑥, 𝑦) − 𝜁]

+_}

22_{Authors express the loss associated to a portfolio in function of percentages vector 𝑥}

and the vector of each asset’s future return 𝑦: the loss is then equal to −𝑥′𝑦.

23_{Acerbi and Tasche (2002b) let us understand that ES and CVaR are essentially two}

different labels employed to identify the same object, thus the expected loss in (1 − 𝛼) of cases.

(33)

33

where [𝑓(𝑥, 𝑦) − 𝜁]+ = max{𝑓(𝑥, 𝑦) − 𝜁; 0}.

From the demonstration of a relevant theorem, thanks to whom the authors proved how it is possible to determine VaRα through a two-step approach24, it

derives that minimization of CVaRα associated to a portfolio 𝑥 is equivalent to

Fα(𝑥, 𝜁) minimization on the entire domain: min_𝑥∈𝑋 CVaRα(𝑥) =_{(𝑥,𝜁)∈𝑋×𝑅}min 𝐹𝛼(𝑥, 𝜁).

This last result is remarkable since, with the aim of defining vector 𝑥 that minimizes CVaRα, it allows to work directly with a simple expression, convex with

respect to variable 𝜁 in 𝐹𝛼(𝑥, 𝜁), rather than managing an expression that requires

the knowledge ex-ante of VaRα’s value. Lastly, the numerical analysis conducted

by Rockafellar and Uryasev (2000) guarantees how such process is implicitly valid also for VaRα minimization, being 𝐶𝑉𝑎𝑅𝛼 ≥ 𝑉𝑎𝑅𝛼.

1.5.3 A new coherent risk measure: Entropic Value-at-Risk

Sometimes being coherent for a risk measure could not be enough; an important deficiency of CVaR, or Expected Shortfall, is that it cannot be computed in a reasonable time. Indeed, in most of the cases is it necessary to approximate CVaR through sampling methods. There are also other examples of coherent risk measures, as spectral risk measures25_{, which cannot be efficiently computed even}

for simple cases. Having to face a stochastic optimization problem, as the portfolio selection one, incorporating a risk measure that has to be computed frequently, makes the need for an efficiently computable coherent risk measure more essential and relevant.

In an important paper of the Journal of Optimization Theory and Applications, A. Ahmadi-Javid (2012) introduced a new coherent risk measure called Entropic

Value-at-Risk (EVaR). It constitutes “the tightest possible upper bound obtained

24_{The approach consists of a first step which requires the definition of a set of values of 𝜁}

which minimize Fα(𝑥, 𝜁) and a second step which identifies its left extreme in case the set

contains more elements. This process results worthless if not interested in VaR’s value.

(34)

34

from Chernoff inequality26_{for the value-at-risk (VaR) as well as the conditional}

value-at-risk (CVaR)” Ahmadi-Javid (2012). In his work Ahmadi-Javid

demonstrated that a large class of stochastic optimization problems that are computationally intractable with CVaR, is efficiently solvable when incorporating EVaR. The dual representation of EVaR is strictly linked to the Kullback-Leibler27

divergence, also known as relative entropy.

Entropic Value-at-Risk owes its name to its connections with Value-at-Risk and relative entropy. We will describe this risk measure more in details and incorporate it in our portfolio selection model in the next chapter.

26_{Chernoff inequality will be explained in the following chapter.} 27_{Kullback-Leibler divergence is a concept that will be discussed later.}

(35)

(36)

36

Chapter 2

Entropic Value-at-Risk application

In this chapter, Entropic Value-at-Risk’s implementation as a risk measure in a portfolio selection problem is discussed. It has been demonstrated [Ahmadi-Javid (2012)] that this measure, proposed by Ahmadi-Javid itself, is able to efficiently solve a broad class of stochastic optimization problems which are instead intractable with CVaR. Indeed, in recent years EVaR has been discussed in many economists and scholars’ studies, often present in papers related to coherent risk measures and portfolio optimization problems28_.

EVaR will be applied to a realistic portfolio selection model, comprehensive of several constraints generally used in fund management practice, proposed by Corazza, Fasano and Gusso (2013).

2.1 Entropic VaR as a risk measure and its properties

Let the risk measure 𝜌 be a function assigning a real value to a random variable 𝑋 ∈ 𝑿 and let 𝑿 be a set of allowable random variables. Then let (Ω, 𝑭, 𝑃) be a probability space where Ω is a set of all simple events, F is a 𝜎-algebra of subsets of Ω and 𝑃 is a probability measure on F. Furthermore, suppose that 𝑳 is the set

28_{Ahmadi-Javid A. (2012c). Application of information-type divergences to constructing}

multiple-priors and variational preferences. In: Proceedings of IEEE International

Symposium on Information Theory, Cambridge, MA, pp. 538-540.

Ahmadi-Javid A. (2012d). Application of entropic value-at-risk in machine learning with

corrupted input data. In: Proceedings of 11th_{International Conference on Information}

Science, Signal Processing and their Applications (ISSPA), Montreal, QC, pp. 1104-1107. Ahmadi-Javid A. and Pichler A. (2017); Pichler A. (2017); Delbaen F. (2018). See bibliography.

(37)

37

of all Borel measurable functions29_{– random variables – 𝑋: Ω → ℝ, and 𝑿 ⊆ 𝑳 is}

a subspace including all real numbers. It is now possible to define the risk measure 𝜌: 𝑿 → ℝ̅ , where ℝ̅ = ℝ ∪ {−∞, +∞} is the extended real line. For 𝑝 ≥ 1 let 𝑳𝑝

be the set of all Borel measurable functions 𝑋: Ω → ℝ for which 𝐸(|𝑋|𝑝) = ∫|𝑋|𝑝𝑑𝑃 < +∞, 𝑳∞ be the set of all bounded Borel measurable functions, 𝑳𝑀 be

the set of all Borel measurable functions 𝑋: Ω → ℝ whose moment-generating function 𝑀𝑋(𝑧) = 𝐸(𝑒𝑧𝑋) exists ∀𝑧 ∈ ℝ, and 𝑳𝑀+ be the set of all Borel

measurable functions 𝑋: Ω → ℝ whose moment-generating function 𝑀𝑋(𝑧) exists

∀𝑧 ≥ 0. Notice that 𝑳∞⊆ 𝑳𝑀 ⊆ 𝑳𝑝, ∀𝑝 ≥ 1.

As mentioned in chapter 1, Entropic VaR can be described as the tightest possible upper bound obtained from Chernoff inequality for the VaR. Chernoff inequality [Chernoff H. (1952)] for any constant 𝑎 and 𝑋 ∈ 𝑳_𝑀+ is as follows:

Pr(𝑋 ≥ 𝑎) ≤ 𝑒−𝑧𝑎𝑀_𝑋(𝑧), ∀𝑧 > 0.

Solving the equation 𝑒−𝑧𝑎𝑀_𝑋(𝑧) = 𝛼 with respect to 𝑎 for 𝛼 ∈ ]0,1], we obtain

𝑎_𝑋(𝛼, 𝑧) ≔ 𝑧−1_{ln (}𝑀𝑋(𝑧)

𝛼 ),

for which we have Pr (𝑋 ≥ 𝑎𝑋(𝛼, 𝑧)) ≤ 𝛼. Indeed, for each 𝑧 > 0, 𝑎𝑋(𝛼, 𝑧) is an

upper bound for 𝑉𝑎𝑅1−𝛼(𝑋). It is possible now to consider the best upper bound

of this type as a new risk measure that bounds 𝑉𝑎𝑅1−𝛼(𝑋) by using exponential

moments.

29_{A map 𝑓: 𝑋 → 𝑌 between two topological spaces is called Borel measurable if 𝑓}−1_(𝐴)

is a Borel set for any open set A. Note that the 𝜎-algebra of Borel sets of X is the smallest 𝜎-algebra containing the open sets. (Borel function: encyclopediaofmath.org).

(38)

38

Definition. Entropic Value-at-Risk (EVaR). The entropic value-at-risk of 𝑋 ∈

𝑳_𝑀+ with confidence level 1 − 𝛼 is defined as follows:

𝐸𝑉𝑎𝑅_1−𝛼(𝑋) ≔ inf

𝑧>0{𝑎𝑋(𝛼, 𝑧)} = inf𝑧>0{𝑧

−1_{ln (}𝑀𝑋(𝑧)

𝛼 )}.

As proved in Ahmadi-Javid (2012a), EVaR is a coherent risk measure. To find its dual representation and its connection to relative entropy, we can proceed as follows:

Theorem. For every coherent risk measure ρ: 𝐋∞ → ℝ with the Fatou property30,

there exists a set of probability measures 𝔍 on (Ω, 𝐅) such that

𝜌(𝑋) = sup

𝑄∈𝔍

𝐸_𝑄(𝑋).

The above equation is known as the dual representation or robust representation of 𝜌. Furthermore, the expression is an additional demonstration that this risk measure is coherent.

Lemma. Donsker-Varadhan Variational Formula. For any 𝑋 ∈ 𝑳∞,

ln 𝐸𝑃(𝑒𝑋) = sup 𝑄≪𝑃

{𝐸𝑄(𝑋) − 𝐷𝐾𝐿(𝑄||𝑃)},

30_{A translation invariant supermodular mapping 𝜙: 𝐿}∞_{→ ℝ is said to satisfy the Fatou}

property if 𝜙(𝑋) ≥ sup 𝜙(𝑋𝑛), for any sequence (𝑋𝑛)𝑛≥1 of functions, uniformly

(39)

39

where 𝐷𝐾𝐿(𝑄||𝑃) ≔ ∫ 𝑑𝑄 𝑑𝑃(ln

𝑑𝑄

𝑑𝑃) 𝑑𝑃 is the relative entropy

31_{of 𝑄 with respect to}

𝑃, or the Kullback-Leibler divergence32_{from 𝑄 to 𝑃.}

Theorem. The dual representation of 𝐸𝑉𝑎𝑅1−𝛼(𝑋) for 𝑋 ∈ 𝑳∞ has the form:

𝐸𝑉𝑎𝑅1−𝛼(𝑋) = sup 𝑄∈𝔍

𝐸𝑄(𝑋),

where 𝔍 = {𝑄 ≪ 𝑃: 𝐷𝐾𝐿(𝑄||𝑃) ≤ − ln 𝛼}.

Entropic value-at-risk is characterized by two other important properties: the former related to another variable with same distribution, the latter linked to the comparison with VaR and CVaR.

Corollary. For 𝑋, 𝑌 ∈ 𝑳_𝑀, 𝐸𝑉𝑎𝑅_1−𝛼(𝑋) = 𝐸𝑉𝑎𝑅_1−𝛼(𝑌) for all 𝛼 ∈ ]0,1] if and only if 𝐹_𝑋(𝑏) = 𝐹_𝑌(𝑏) for all 𝑏 ∈ ℝ.

The proof of this property follows from a well-known property of moment-generating functions, stating that two distributions are identical if the have the same moment-generating function. This corollary shows that 𝐸𝑉𝑎𝑅1−𝛼(𝑋) as a

function of its parameter 𝛼 characterizes the distribution of 𝑋 ∈ 𝑳𝑀. The initial

condition 𝑋, 𝑌 ∈ 𝑳𝑀 can be weakened to the existence of 𝑀𝑋(𝑏) and 𝑀𝑌(𝑏) over

the interval 𝑏 ∈ ] − 𝜀, +∞] for a positive constant 𝜀 > 0.

31_{Entropy is a concept that derives from physics and allows to evaluate the level of}

disorder in a system. If the level of disorder grows, entropy increases as well, vice versa if it reduces, entropy decreases. Recently this measure has been reproposed in different fields as information theory, IT, biology, medicine and social sciences. In economics and finance entropy has been mostly employed as a measure of risk or as foundation of a more complex risk measure.

32_{Kullback-Leibler divergence – also known as information divergence or relative entropy}

– is a non-symmetric measure of the difference between two probability (P and Q) distributions. More specifically, K-L divergence from Q to P, identified as 𝐷𝐾𝐿(𝑄||𝑃), is the

(40)

40

Proposition. The EVaR is an upper bound for both VaR and CVaR with the same

confidence levels, i.e. for 𝑋 ∈ 𝑳_𝑀+ and every 𝛼 ∈ ]0,1]:

𝐶𝑉𝑎𝑅_1−𝛼(𝑋) ≤ 𝐸𝑉𝑎𝑅_1−𝛼(𝑋). Furthermore, 𝐸(𝑋) ≤ 𝐸𝑉𝑎𝑅_1−𝛼(𝑋) ≤ 𝑒𝑠𝑠 sup(𝑋), where: ▪ 𝐸(𝑋) = 𝐸𝑉𝑎𝑅₀(𝑋); ▪ 𝑒𝑠𝑠 sup(𝑋) = lim 𝛼→0𝐸𝑉𝑎𝑅1−𝛼(𝑋).

This statement affirms that EVaR is more risk-averse with respect to CVaR at the same confidence level. Thus, EVaR would suggest to a financial firm allocating more resources to avoid risk. This feature could make EVaR less attractive for companies which search, for instance, a greater return and are not afraid of risk; however, EVaR computational tractability results more simple, which can be important when we need to incorporate a risk measure in a stochastic optimization problem, both in terms of time and difficulty.

2.2 A realistic portfolio selection model under EVaR

Once an appropriate measure of risk is identified, with the purpose of quantifying the level of riskiness inherent in a financial investment, there is the necessity to introduce a set of constraints in order to develop a realistic portfolio selection model.

Even though it could be taken for granted, it is important to underline how the research of a portfolio which is able to minimize any measure of risk, being it represented by returns’ variance or a coherent risk measure, does not lead to a solution that can be adopted in practical terms if not supplied by constraints to be

(41)

41

followed. Referring to what described in chapter 1, the generic rational investor has to deal with the choice between conflicting objectives as return maximization and risk minimization, in the overall search of expected utility maximization. Risk minimization without taking into consideration constraints in terms of expected return, but considering only the budget constraint33_{, would lead to an}

optimal solution that is given by the minimum risk portfolio34_{. Without explaining}

in details logics and reasonings of expected utility theory, since they are not part of the aim of our work, it appears clear that this efficient solution is only one among many admissible and does not take into account the level of risk aversion of each investor, transmitted through a specific utility function.

Even considering the constraint related to the expected return, in addition to the pre-mentioned budget constraint, the solution appears too simplistic with respect to the reality of financial markets. For this reason, the model proposed in this work will take into account transaction costs, in other words those costs that the intermediary will charge to the investor once the transaction in completed and that represent, together with the tax regime, a significant financial market friction. Transaction costs will be considered indirectly through the introduction of cardinality constraints. The author believes it is appropriate to add these constraints to our model since they represent one of the major constraints categories to which a manager is subjected in daily practice.

Nevertheless, it is important to highlight that each time that a constraint is added to the model, its computational complexity increases proportionally and even more rapidly when the added constraints do not have linear and/or continuous forms. The problems that generate after the insertion of such constraints are called NP-hard, which, due to their difficulty in terms of computation and time, require the employment of heuristics and metaheuristics to be solved, since the

33_{Risk unconstrained minimization would inevitably lead to a vector of zero percentages}

of investment in each asset, since no portion of wealth would be allocated in an asset with an aleatory return.

34_{In the Markowitz’s mean-variance approach it is called “global minimum variance}

(42)

42

research of optimum solutions through the use of exact methods might not lead to a result.

Before the final proposal of the realistic portfolio selection model, it is important to list and describe in detail the constraints that it has been decided to consider in the model.

2.2.1 Budget and return constraints

Budget constraint and return constraint are widespread in most of portfolio selection models, since they represent the essential part of the problem. They are also included in the constrained Markowitz model.

Budget constraint assures that all available wealth is invested and can be

mathematically formulated as follows:

∑ 𝑥_𝑖

𝑁

𝑖=1

= 1

where 𝑥𝑖 represents the percentage of capital invested in the 𝑖 − 𝑡ℎ asset. Budget

constraint can also be expressed in matrix terms as:

𝑥′𝑒 = 1.

Return constraint establishes the determination, on the investor side, of a certain

level of desired return 𝜋 and guarantees that the expected return of the portfolio is not below that threshold. Hence, it permits to select a portfolio among all portfolios lying on the efficient frontier: the one which minimizes the adopted measure of risk, in our case the entropic value-at-risk. In algebric terms, portfolio

(43)

43

mean return is equal to 𝜇𝑝= ∑ 𝑥𝑁𝑖 𝑖𝑟𝑖, whereas in matrix terms it is defined by 𝑥′𝑟.

The constraint on the minimum mean return can be formulated as follows:

∑ 𝑥_𝑖𝑟_𝑖

𝑁

𝑖=1

≥ 𝜋.

In the specific problem that is being proposed, the perfect equality between mean return and 𝜋 is not required, whereas a wider condition is imposed in order to avoid that the admissible region results empty. Indeed, any rational investor would not be dissatisfied by a portfolio that, at the same risk level, provides a return greater than expected.

From a conceptual point of view, it is possible to reach the same result if deciding to maximize return once a certain level of risk is fixed. However, considering the problem from a risk-averse investor’s perspective, the concept underlying the choice to configurate the objective function that must be optimized with a measure of risk – to be minimized – instead of a measure of the return – to be maximized – reflects the investor’s aversion to risk; investor who tends to focus his attention on risk rather than on return. This occurs notwithstanding these two variables are positively correlated. An example of this tendency towards risk is represented by the so-called MIFID35_{interview, which is applied by credit}

institutions to their clients in order to verify both their knowledge and experience referring to products and financial tools and their investment goals. Questions that constitutes the interview, deal with the risk aversion of the investor rather than with its propension to profit. Hence, it is possible to obtain a financial profile of

35_{The 2004/39/EU directive MIFID (Markets in Financial Instruments Directive) was}

introduced in 2007 and has the objective of increasing investor protection and guaranteeing the maximum level of transparency through mandatory information to customers.

(44)

44

the client which allows to guide him through the formulation of an adequate investment choice.

2.2.2 Cardinality constraints

Cardinality constraints are bounds related to the number of assets to include in the portfolio; they will also be associated with a constraint regarding the fraction of portfolio that each asset constitutes. The reason for the introduction of these constraints is to obtain an indirect control on transaction costs. The majority of models presented in literature considers, especially in the case in which the focus is on the substance of the approach rather than on the form, a system of constraints rather elementary, based on the assumption of perfect market conditions, frictionless. However, transaction costs, in addition to being significant incidence factors on the effective real performances of the portfolio, represent a constraints category that needs to be taken into consideration in the development of a portfolio model since their presence considerably influences managers activities in their daily practice.

Since the trade of assets comes with a cost, portfolio managers are led to subordinate portfolio creation to costs that has to be borne. For this reason, it is important that they operate, in first place, introducing a not too small not too big number of assets among the N constituting the basket of financial tools at their disposal. In fact, when the amount of assets selected is too large, many practical issues can befall, e.g. high dimensionality of the problem which can raise transaction costs as well. In second place, portfolio managers will also be subordinated to the selection of an investment percentage for each asset that must be not too small and not too big; these fractions of portfolio will be strictly correlated with the minimum and maximum number of assets present in the portfolio.

We can introduce and consider transaction costs in our portfolio selection problem through the employment of the following cardinality constraint: